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Our Universe
Infinite and Eternal
—
Its Physics, Nature, and Cosmology
Barry Bruce
Universal-Publishers
Boca Raton
Our Universe—Infinite and Eternal: Its Physics, Nature, and Cosmology
c 2012 Barry Bruce
Copyright All rights reserved.
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Universal Publishers
Boca Raton, Florida USA
2012
ISBN-10: 1-61233-160-2
ISBN-13: 978-1-61233-160-7
www.universal-publishers.com
Publisher’s Cataloging-in-Publication Data
Bruce, Barry.
Our universe : infinite and eternal : its physics, nature, and
cosmology / Barry Bruce.
p. cm.
Includes bibliographical references and index.
ISBN: 978-1-61233-160-7
1. General relativity (Physics)Mathematics. 2. Gravitation.
3. Cosmology. 4. Quantum theory. I. Title.
QC173.6 .B78 2012
530.11dc23
2012949298
To Newton and Einstein
To all the teachers I have had along the way
To all my friends who have helped me make it sanely this far into
life
To my brother and sister who have given me a sense of family
To my parents who gave me life and raised me to be the person I
have become today
Preface
The first part of the book deals with the foundations and nature
of physics. It will be shown that underlying basic physical laws of
Newton are likely the result of the distribution of matter within
our Universe and the gravitational force between matter.
In the first chapter of the book, the field equations for Einstein’s
general relativity are solved for a static infinite universe that has,
on average, a uniform mass density. One of the three solutions for
this uniform density, infinite universe fits the Hubble diagram red
shift data better than the Big Bang. The red shift predictions for
this solution fits the Hubble straight line data. In addition, the
predictions of this model fits the high red shift supernova data as
a natural consequence of the uniform distribution of matter within
an infinite universe without the need of postulating an unknown
dark energy. Moreover, this solution fits, the all but unknown,
nearby galaxy red shift data of Sandage and Visvanathan from the
late 1970’s. That is, this conceptually simple model of the Universe
fits all the presently observed red shift versus magnitude data.
In the third chapter, we develop the relativistically correct force
of gravity using the general relativistic Schwarzschild metric for a
non-rotating massive point particle, and the Lagrangian formulation of classical mechanics. We then use this force of gravity to
calculate the gravitational force on a particle in arbitrary motion
due to all the matter in this static, uniform density, infinite univ
vi
PREFACE
verse. The force of gravity so calculated turns out to be equal to
the Force of Inertia. The results of this calculations show that the
origin of Newton’s First Law and Newton’s Second Law is due to
gravity and the uniform distribution of matter in an infinite Universe. The calculation also shows the inertial mass is equal to the
gravitational mass.
We call this universe that fits all of the red shift data, unites
the physics of Newton with the physics of Einstein, and agrees
with both of their intuitions about the nature of the universe, the
Infinite Universe of Einstein and Newton. Because the Infinite
Universe of Einstein and Newton fits the observational data and
reproduces the basic physics of the Universe that we have come to
know, it is quite likely that our Universe is eternal and infinite.
The discussion in chapter 4 explores some qualitative implications for the nature of our Universe if the Infinite Universe of
Einstein and Newton constitutes an appropriate model for our Universe. One result that is required by the nature of the solution for
the Infinite Universe of Einstein and Newton is that each point
in space-time is causally connected only to a finite subuniverse of
the whole infinite universe. However, because each point in spacetime has its own unique subuniverse to which it has its own causal
connection, a small random component is introduced into the dynamics of every object. This likely forms the physical basis for the
Uncertainty Principle.
In chapter 5, we show that the advanced forms of classical mechanics, physics that is consistent with the basic physics of Newton
and Einstein, and thus consistent with the physics of the Infinite
Universe of Einstein and Newton, leads to the result that associated with each particle is the phase of a wave, which has the speed
of the de Broglie wave. We also develop the form of the phase of
the de Broglie wave associated with a particle.
In chapter 6, we develop quantum mechanics from the basic
physics consistent with the Infinite Universe of Einstein and New-
vii
ton. To do this we develop the Schrödinger equation by solving the
Hamilton-Jacobi equation, the abstract particle equation of classical mechanics, using an integrating factor. We form a wave for
the particle by creating the complex exponential of the phase we
developed for the de Broglie wave. We use this functional form
as the integrating factor in the Hamilton-Jacobi equation. Easy
manipulation then allows us to put this Hamilton-Jacobi differential equation into the form of the Schrödinger operator equation of
quantum mechanics.
We then move on to show that the classical wave function has
all the mathematical properties of the quantum mechanical wave
function. This includes the fact that the absolute valued squared
of the normalized classical wave function has the mathematical
properties of a probability density. However, because a particle
need never go where its associated wave goes, such a probability
density property does not indicate the likelihood of finding the
particle in a particular vicinity of space.
What we have shown by our development is that quantum
mechanics is consistent with and expected to obtain in the Infinite Universe of Einstein and Newton. Our development has also
shown that a solution to the particle Hamilton-Jacobi equation exists whenever a solution to the quantum mechanical wave equation
exists and vice versa. That is, the wave-particle duality is a reality
in this Universe.
Part II deals with topics of a different tone. The discussion
does not deal directly with the foundations of physics, but instead
deals with topics that are associated with the nature of our Universe. The topics discussed will not lead to the development of
new physics. The topics presented instead will use physics to help
develop an understanding about the nature of our Universe. These
topics tend to involve the more complex interactions of several
physical entities and thus the discussion tends to be more involved
than the discussions presented in Part I. Readers not familiar with
viii
PREFACE
the nitty gritty of a topic might find a first reading difficult. However, the first reading should give the reader the essentials of the
needed background so that a second reading should proceed more
easily and give a fuller understanding of what is being presented.
The only topic so far in this part of the book is the missing
matter problem of astrophysics, which appears in chapter 8. To
attack this problem we consider the dynamics of spiral galaxies.
We model a spiral galaxy simply. We model it as a spherical central
bulge surrounded by a thin flat disk of stars and gas that revolves
about the central bulge. Surrounding both the disk and the bulge is
a diffuse spherical halo. The calculation of the dynamics resulting
from the gravitation of the thin disk is the only complication of the
problem. It turns out that we need to numerically integrate one
dimension of the three dimensional integral in order to explore the
dynamics. But other than that the problem is straight forward.
In our first iteration of the problem, we consider the dynamics
of our Milky Way—the galaxy we know best. The calculations
allow an easy consistent fit to the data.
After completing the calculations for the Milky Way, I discovered the existence of data for neighboring galaxies that showed the
existence of a gaseous disk that extends two to three times further from the central bulge than a stellar disk does. Moreover, the
gaseous disk does so with the same high rate of rotation as occurs
with a stellar disk. Calculations made in the same manner as those
that were used for the stellar disk in the Milky Way produced a
reasonable fit. However, the fit could be better.
A gaseous disk is seen by the radiation of the 21-cm line resulting from the spin flip in hydrogen gas. A search of the literature
produced a range of mass densities in gas clouds of hydrogen from
which stars are born and a more diffuse range of mass densities that
have resulted in the observation of relatively strong radiation from
the 21-cm line of hydrogen. These density properties led to the
hypothesis that the transition from the stellar disk to the gaseous
ix
disk would occur when the density of a galactic disk decays away
to a value close to that of the lower limit of the range of mass densities in which stellar formation has been observed. Furthermore,
it is hypothesized that a gaseous disk is observed until the density
of the disk decays away to a value in the lower part of the range
in which a relatively strong 21-cm line emission has been observed.
After calculating a model that fit rotation speeds of the stellar disk
and rotation speeds of the gaseous disk in a particularly well resolved nearby spiral galaxy, it was found that mass densities of the
disk so modeled fit well with these two hypotheses.
A small tweak was made and added to the calculation when I realized that gas would likely cohere together more than stars would.
Accordingly, the decay length for the gaseous disk was lengthened
to a value slightly greater than that used for the stellar disk. Making this change noticeably improved the fit to the rotational speeds
of the well resolved nearby spiral galaxy.
While researching the density of hydrogen gas in galaxies, I
came across work which observed that in the halo of the Milky
Way, far from the center of the galaxy, there is hot coronal gas
at low density. This led to the hypothesis that gas in this low
density and high temperature region formed a natural boundary
for the outer edge of a spiral galaxy. That is, in this region of low
density and high temperature, the gas of a galaxy’s halo slowly
boils off like the gas in a planet’s atmosphere that is near to the
planet’s Sun. When the halo of the model for the well resolved
neighboring galaxy was extended out until this low density was
reached, it was found that the gas of its halo would boil off slowly
if the temperature in this low density edge of the halo possessed
a temperature similar to that of the hot coronal gas in the Milky
Way. This gives mild support to the hypothesis.
The above hypotheses were then applied to model the Milky
Way. It was found that the Milky Way, so modeled with ordinary
matter, could be made to fit all the dynamic data well. In addition,
x
PREFACE
it is shown that the size and distribution of matter is consistent
with a resolution of the missing matter problem.
In the last part of chapter 8, our attention turns to making
the amount of non-luminous matter in the disk seem reasonable
relative to the amount of luminous matter. Our analysis centers
on the mass and distribution of the very small dim stars and brown
dwarfs along with stellar remnants; black holes, neutron stars, and
white dwarfs. In this discussion, we make use of the initial mass
function. This analysis results in the expectation that there is
enough ordinary matter in the disk, taken together with the mass
of the bulge and the halo, to account for the properties of the
orbital dynamics of the stars in the disk.
I have only included topics in the present book on which I
thought I had done enough preliminary work so that I would be
able to work out whatever else was needed as I wrote. I have other
ideas in mind, but at my age one never knows how much more time
one will have to work on them. Thus, I view my present endeavor
as a work in progress. New results, as I develop them, will appear
in a fresh edition of this book.
Barry Bruce
August 2012
Contents
Preface
v
I
1
On the Foundation and Nature of Physics
1 The Infinite Universe
1.1 Physical Ideas . . . . . . . . . . . . . . . . .
1.1.1 The Infinite Limit Conception . . . .
1.1.2 Classical Analysis . . . . . . . . . .
1.2 General Relativistic Analysis . . . . . . . .
1.2.1 Application of Boundary Conditions
1.2.2 Nature of the Solutions . . . . . . .
1.3 Developing the Cosmological Redshift
Expressions . . . . . . . . . . . . . . . . . .
1.3.1 Infinite Open Universe . . . . . . . .
1.3.2 Infinite Flat Universe . . . . . . . .
1.3.3 Infinite Closed Universe . . . . . . .
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2 Comparing Models of the Universe
29
2.1 Qualitative Discussion . . . . . . . . . . . . . . . . . 29
2.1.1 Quasars . . . . . . . . . . . . . . . . . . . . . 32
2.2 Hubble’s Law . . . . . . . . . . . . . . . . . . . . . . 36
xi
xii
Contents
The z vs m relation
Universe . . . . . . .
2.2.2 Hubble Diagrams . .
2.3 Conclusions . . . . . . . . .
2.2.1
for
. .
. .
. .
the Infinite
. . . . . . .
. . . . . . .
. . . . . . .
Closed
. . . . . 37
. . . . . 39
. . . . . 46
3 Newton’s Laws and the Force of Gravity
3.1 Introductory Ideas . . . . . . . . . . . . . . . . .
3.2 The Force of Gravity . . . . . . . . . . . . . . . .
3.3 Mach’s Principle and Newton’s Laws . . . . . . .
3.3.1 Newton’s Laws . . . . . . . . . . . . . . .
3.4 Newton’s Third Law . . . . . . . . . . . . . . . .
3.4.1 A Note on the Reduced Mass Formalism.
3.4.2 A Note on the Nature of Metrics . . . . .
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4 Discussion
4.1 Foundational Heuristics . . . . . . . . . . . . . . . .
4.2 Einstein and Mach’s Principle . . . . . . . . . . . . .
4.2.1 Cosmological Implication of General Relativity
4.3 The Universe of Einstein and Newton . . . . . . . .
4.3.1 Newton’s Ideas concerning Space and Inertia
4.3.2 Frames of Reference . . . . . . . . . . . . . .
4.3.3 Universe of Subuniverses . . . . . . . . . . . .
4.3.4 Time and Space-time . . . . . . . . . . . . .
4.3.5 The Seeds of quantum mechanics . . . . . . .
5 Wave Particle Duality
5.1 Introduction . . . . . . . . . . . . . . .
5.2 Physics in the Laboratory . . . . . . .
5.2.1 Physics of Galaxies and Galaxy
5.3 Classical Equations of Motion . . . . .
5.4 de Broglie Waves . . . . . . . . . . . .
5.4.1 A Note on Wave Speed . . . .
5.5 Lagrangian Connection . . . . . . . . .
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Clusters
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91
94
95
97
97
98
100
101
105
114
115
Contents
xiii
6 Quantum Mechanics
117
6.1 The Schrödinger Equation . . . . . . . . . . . . . . . 118
6.1.1 Commutators . . . . . . . . . . . . . . . . . . 124
6.2 The Wave Function . . . . . . . . . . . . . . . . . . . 125
II
Other Topics on the Nature of the Universe
137
7 Spiral Galaxies and Dark Matter
139
7.1 Overview to Our Analysis . . . . . . . . . . . . . . . 140
7.2 Analysis of Vertical Motion in a Spiral Galaxy’s Disk 143
7.2.1 Lower Bound for Disk Matter Density . . . . 145
7.2.2 Upper Bound for Disk Matter Density . . . . 150
7.3 Disk Rotational Speed . . . . . . . . . . . . . . . . . 155
7.3.1 The Central Bulge . . . . . . . . . . . . . . . 157
7.3.2 The Halo . . . . . . . . . . . . . . . . . . . . 160
7.3.3 The Disk . . . . . . . . . . . . . . . . . . . . 163
7.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . 178
7.5 Nearby Disk Galaxies . . . . . . . . . . . . . . . . . 180
7.5.1 Conjectures . . . . . . . . . . . . . . . . . . . 184
7.6 The Extended Milky Way . . . . . . . . . . . . . . . 188
7.6.1 Are the Magellanic Clouds Satellite Galaxies? 191
7.7 Disk Dark Matter . . . . . . . . . . . . . . . . . . . . 198
7.7.1 Spiral Arms . . . . . . . . . . . . . . . . . . . 201
7.7.2 Vertical Disk Density . . . . . . . . . . . . . 203
7.7.3 Dark Stellar Remnants . . . . . . . . . . . . . 209
7.7.4 Brown Dwarfs and Very Dim Stars . . . . . . 214
7.7.5 Conclusions . . . . . . . . . . . . . . . . . . . 227
A Maximal Redshift of the Infinite Closed Universe 229
Bibliography
239
Index
249
Part I
On the Foundation and
Nature of Physics
1
Chapter 1
The Infinite Universe
This chapter presents physical ideas grounded in the observed isotropic homogeneous nature of the Universe that lead to the conclusion that the Universe might be infinite. Following this possibility,
three metrics for an infinite, static, isotropic, and homogeneous
universe are found, which satisfy the field equations of Einstein’s
general relativity. Two of these metrics show that light traveling
in their universe would be gravitationally red shifted. In addition,
the universe of one of these two metrics, the infinite closed universe, would trap light in a finite portion of the universe, so that
each point in space would have its own finite universe. The second
chapter concludes by showing that the infinite closed universe fits
the data of the Hubble diagram better than the Big Bang Theory.
1.1
Physical Ideas
There are three fundamental observational facts about the nature
of our Universe. One is that the Universe on a very large scale
seems, on average, to be homogeneous and isotropic. The second
observation is that the light coming from far away objects is red
shifted. The further the object is away from us, the larger on
3
4
Chapter 1
The Infinite Universe
average is the redshift. The third fact is that the night sky is
dark. This last observation presents a problem—known as Olbers’
paradox: “If the Universe is infinite with a finite number of stars
per unit volume then the night sky would be expected to have a
brightness equal to the brightness of the surface of an average star.”
The currently held view is that the Universe is finite and expanding everywhere and that this expansion is the cause of the
cosmological redshift. Every one acknowledges the existence of
gravitational red shifting. However, it is thought that gravitational red shifting effects are much too small to account for the
observed cosmological redshift. But are they? The Schwarzschild
line element (Adler et al. 1975, d’Inverno 1992, Weinberg 1972)
for a spherically symmetric static massive “object” exterior to the
object is
2
ds
=
2GM
1− 2
c r
1
dr 2
2GM
1− 2
c r
−r 2 dθ2 − r 2 sin2 θdφ2 .
c2 dt2 −
(1.1)
The mass M for any spherical
object of constant density ρ is M =
4
3c2
3
3 πρr . So that when r =
8πGρ , grr in the Schwarzschild metric
becomes infinite and gtt equals zero. This means that no matter
how small the density, there is a radius for an object of that density
for which it would be a black hole. Consequently, light emanating
from near the object’s center would be severely redshifted as it
approached the boundary of the object.
How big would our Universe be if it were a black hole so
that no light could pass out of it? If we assume a density of
ρ = 1 × 10−27 kg m−3 , which is about half the currently accepted
best value, the Schwarzschild radius of our Universe would be about
42 billion light years. This is about 1.4 times the presently accepted
value of the radius of the universe “now”. If one used the current
best value of the density of the universe, one would get a Schwarz-
1.1
Physical Ideas
5
schild radius for our Universe about equal to the currently accepted
value for the radius of our Universe. However, this is not the point.
Whatever else this tells us, it is clear that in a finite universe gravitational shifting need not be negligible and that any light from a
very distant source would in general have its wavelength severely
shifted due to gravity.
Light traveling through a finite diffusely filled universe, no matter how large, is similar to a ball traveling through a smooth frictionless tunnel carved inside the Earth. A ball in such a tunnel will
pick up speed (kinetic energy) as it comes closer to the center of the
Earth and it will slow down, that is, lose kinetic energy as it gets
farther from the center. Similarly, light would lose energy, that is,
its frequency, f = E/h, would decrease, as the light gets further
from the center of a finite universe. Conversely, the frequency of
light would increase as light gets closer to the center of such a universe. Put in terms of wavelength, if the source of the light were
closer to the center of the universe than an observer, the observer
would observe the light as red shifted. However, if the source of the
light were further from the center of the universe than the observer
then the light would appear blue shifted. If our Universe were an
expanding finite universe, the light would be redshifted because of
the expansion. However, because of the size of the gravitational
shifting of light, there would be a clear and undeniable asymmetry
in the observed redshift for any observer not situated at the center
of the Universe. Since this is not what we observe, this suggests
that the Universe may be infinite. The Big Bang Universe assumes
a different nature for the geometry of a finite universe in order to
circumvent this problem. We, however, will pursue the possibility
of an infinite universe with ordinary type geometry.
1.1.1
The Infinite Limit Conception
As I was taught and as one teaches in freshman physics, the gravitational force on a material particle in a region which has a fi-
6
Chapter 1
The Infinite Universe
nite spherically symmetric mass density, no matter how large the
sphere, is toward the center of the sphere and has a magnitude that
is only dependent on the matter which is closer to the center than
the particle. Further, the net force on the particle depends not
a whit on the amount of matter which is farther from the center
than it. In addition, the potential energy between the particle and
mass distribution increases as the particle gets further and further
from the center. If one considers an isotropic homogeneous infinite
universe with uniform non-zero density to be the limit of a finite
spherical universe, which keeps its density fixed as r → ∞, then
this potential energy property would hold for the infinite universe
too.
In a homogeneous, isotropic, infinite universe each point in
space is equivalent to any other point on the cosmological scale.
That is, each point in space can be considered an origin about
which gravitating “matter” is spherically symmetric in the cosmological sense. Then, in the infinite limit conception, as a light ray
“pulls away” from its source, the origin of its coordinates, which
corresponds to the same natural origin of a spherical wave, more
and more gravitating mass lies interior to it. This means the light’s
potential energy rises and thus its frequency, i.e., its energy decreases. So, in an isotropic, homogeneous, infinite universe, light is
reddened the further it gets from its originating source. In addition,
since the universe is homogeneous, it is reddened on average by the
same amount for all light which has traveled the same distance.
This means an observer will see light which has been red shifted
coming at him from all directions. The light will be red shifted
more, the further away the source. Furthermore, the isotropy of
such an infinite universe would mean that the red shifting of observed light would be, in the large, independent of direction.
This conception of light pulling away from its source point and
“seeing” more and more gravitating mass in an homogeneous infinite universe gives impetus and focus to what is to follow. However,
1.1
Physical Ideas
7
this conception connected with infinite space might seem somewhat
counter-intuitive because one might expect the “pull” from one side
to cancel the “pull” from the opposite side, and thus there might
be no net effect and the space would be flat. Nevertheless, we
proceed. (It turns out that later we will see a resolution to these
conflicting symmetries.)
1.1.2
Classical Analysis
In this section we will analyze the static homogeneous, isotropic,
infinite universe classically, using the infinite limit conception, in
order to test whether it matches our qualitative reasoning above.
To do this we consider two concentric spherically symmetric regions
of constant density ρ1 and ρ2 as shown in Fig. 1.1.
R1
ρ1
ρ2
Figure 1.1
variables.
r
Two concentric spherically symmetric regions used to help define
8
Chapter 1
The Infinite Universe
In region 1, the force, F1 on a particle of mass, m, is equal
to −GMin m/r 2 r̂ where Min = 43 πρ1 r 3 . Since F1 = −∇U1 , the
potential energy associated with the particle in region 1 is given by
U1 =
2πρ1 Gmr 2
+ k1 .
3
Similarly, for a particle of mass, m, in region 2 the potential energy
is given by
U2 = −
4π 3
3 R1 (ρ1
− ρ2 )Gm 2π
+
Gρ2 mr 2 + k2 .
r
3
For a finite universe of radius R1 (ρ2 = 0) one would set k2 = 0,
so that the gravitational potential energy is zero at infinity. However, for an infinite universe one takes the gravitational potential energy, zero reference level at the origin, by setting k1 = 0.
Then, requiring continuity of the potential energy at R1 gives
k2 = 2πGm(ρ1 − ρ2 )R21 , which results in
U1 =
2π
Gmρ1 r 2
3
(1.2a)
and
U2 = 2πGm(ρ1 − ρ2 )R21 −
3
Gm 4π
2π
3 (ρ1 − ρ2 )R1
+ Gmρ2r 2 . (1.2b)
r
3
For the infinite universe, let us consider the simple idealized
case in which light is emitted from some atom in space (the origin)
and not from the surface of a star. To do this we set ρ1 = ρ2 = ρ.
Then, the potential energy is
U=
2π
Gmρr 2
3
(1.3a)
or for later use, the potential is given by
V =
2πGρr 2
.
3
(1.3b)
1.1
Physical Ideas
9
We apply conservation of energy, utilizing Eq. (1.3a), and set the
energy of the light at the source equal to the energy of the light at
the observer. This gives
hf0 = hf +
2π hf 2
G 2 ρr ,
3
c
(1.4)
where hf
c2 has been used for the “mass” of the photon. This means
that classically, the frequency of the light seen by the observer, in
terms of the frequency of the light emitted by the source, is
f=
f0
.
2πGρ 2
1+
r
3c2
In other words, the wavelength of the observed light is reddened
with an observed wavelength given by
λ= 1+
2πGρ 2 r λ0 .
3c2
(1.5)
Note, that in this analysis the light is not trapped in a black
hole since the energy of a photon is only reduced to zero in the limit
as r approaches infinity. Thus, this analysis is not fully consistent
with general relativity, since no black hole is manifest. However,
the analysis does show that light is red shifted as it moves away
from its source.
If the light leaves the surface of a star of radius R1 then conservation of energy results in the equation
hf0 +
2π hf0 2π hf G 2 ρstR21 = hf +
G 2 ρsp r 2
3
c
3
c
4π
3
G( hf
)
(ρ
st − ρsp )R1
c2 3
−
r
hf + 2πG 2 (ρst − ρsp)R21 .
c
10
Chapter 1
The Infinite Universe
Assuming that the density of space ρsp is negligible compared to
ρst of the star and that the observer is far from the star so that
the 1/r term can be neglected, the expression for conservation of
energy reduces to
2π hf0 2π hf G 2 ρstR21 = hf +
G 2 ρsp r 2
3
c
3
c
hf + 2πG 2 ρst R21 .
c
Thus, the frequency of the observed light is given by
hf0 +
2πGρstR21
3c2
f0 .
f=
2
2πGρstR1
2π
2
1+
+ 2 Gρsp r
c2
3c
Here, we see that light from a star is red shifted similar to light
emanating from empty space. However, as before we see that no
black hole is manifest, no matter how massive the star. Since,
recent observations have made it seem very likely that there are
black holes at the center of many galaxies (Genzel et al. 1997, Ghez
et al. 1998) and have also indicated that there are black hole double
“stars” in which one of the massive objects is a black hole (Casares
and Charles 1994, Shahbaz et al. 1994), the existence of black holes
is all but confirmed. Thus, the results above indicate the need to
do a full general relativistic analysis whenever the gravitational
effects are very large.
1+
1.2
General Relativistic Analysis
In order to proceed with the general relativistic analysis, we will
take a metric of a most general form which satisfies the conditions
which are observed and also satisfies the infinite universe hypothesis. We want our metric to represent a universe which is homogeneous, isotropic and static in the cosmological sense about any
1.2
General Relativistic Analysis
11
point in the Universe. In order to isolate the cosmological properties of the Universe from the properties due to local irregularities,
we will consider that all the matter in the Universe is spread out
into a uniform matter density.
Because we are considering the universe to be static, time becomes irrelevant, that is, all times are equivalent. This is so because
in a static universe, from a given position in space, the universe
would always present similar physics in the cosmological sense no
matter when observations are made. Thus, the homogeneity we are
assuming allows observations by various observers, which can be
made at any time, and only requires that observations of different
observers “look” the same in the cosmological sense. Thus, it is not
necessary that all observers be able to coordinate their times. So,
in order to keep our assumptions to a minimum, we do not make
the common cosmological assumption concerning the existence of
a global Gaussian coordinate system. In particular, we do not assume that there exists a global time coordinate, x0 = ct, such that
the Gaussian metric
ds2 = c2 dt2 + gij dxi dxj
i, j = 1, 2, 3
(1.6)
exists globally. Instead, we let the nature of time come out as it
will. In this way we let the cosmological properties of our model
be the result of its matter distribution and gravitation alone.
The development of the Schwarzschild metric (Adler et al. 1975,
d’Inverno 1992, Weinberg 1972) begins with writing a canonical
general form for a metric which is static and isotropic about a
particular point—the origin. The origin might be the center of a
star or black hole. Or, it might be nothing at all. The general
metric chosen in the Schwarzschild development is defined by the
line element
ds2 = eν(r) c2 dt2 − eλ(r)dr 2 − r 2 dθ2 − r 2 sin2 θdφ2 .
(1.7)